probably goes without saying but take care when using advice from an internet forum.

That said, my own advice for what it's worth, would be for you to contact the Citizens advice bureau, they have very detailed guidelines for settling neighborly disputes and can assist with mediation. even if they cannot help settle the matter of payment for damages, contacting them and attempting mediation could look good on paper for your side of the argument should you have to go to court to seek remuneration. However be warned that in neighbor disputes of this nature, the citizens advice bureau will assess whether or not you need to do something to reduce the impact your dogs noise makes on your neighbor, you might find yourself incurring costs to build a higher fence, which may or may not be the responsibility of the landlord, who may then seek to invoke a no pets clause in the lease rather than go to the effort and expense of any soundproof measures deemed necessary.

Make sure you go into any option you decide to pursue with your eyes open. If you're sure he's in the wrong and over-reacting to a freak occurrence, then you might find that such anti-social behaviour and potential cruelty to animals (especially if he was trying to shut the dogs up by causing them some kind of injury) could land him in a whole world of trouble, and if an independent mediator was to point out how much trouble he could be in, he might think that buying a new microwave to keep you quiet the sensible choice.